Perhaps in this fanfiction, Naruto, abandoned by Kushina, ends up in a world where ninjas don't exist in the same way. He has to adapt to a completely different lifestyle. But he still holds onto his ninja spirit. As he grows, he discovers secrets about his family that Kushina never told him and that knowledge helps him become stronger and more self - assured.
In a Naruto and One Piece crossover fanfiction with this theme, Naruto could be found by the Straw Hat Pirates after being abandoned. Luffy might see Naruto's strength and determination and invite him to join the crew. Naruto would then have to navigate the Grand Line, using his ninja abilities in a world full of pirates. It could also lead to some interesting battles where Naruto's jutsu combines with the Devil Fruit powers of the crew members.
Well, it's hard to specifically name a particular 'Naruto abandoned crossover fanfiction' as there are so many out there. However, often when an author tries to cross Naruto with a Western - style fantasy world, they might abandon the story. Maybe they realize later that integrating the Naruto universe's ninja system with a completely different magic system is more difficult than they thought, and they just give up halfway through writing the fanfiction.
The existence of Naruto abandoned crossover fanfictions can also be due to the difficulty of maintaining consistency. In a crossover, not only do you have to keep the Naruto universe rules intact, but also those of the other universe. For example, in a Naruto and 'Fullmetal Alchemist' crossover, the author has to make sure that the laws of equivalent exchange in 'Fullmetal Alchemist' don't conflict with Naruto's power - up systems. If it becomes too much of a headache to keep everything in order, the author may just drop the fanfiction.
One character could be the person who finds Naruto after being abandoned. It could be a powerful and kind - hearted samurai from a different world. This samurai takes Naruto under his wing and teaches him about honor and different combat techniques.
Naruto's character would likely become more guarded at first. In a crossover with the Attack on Titan world, for example, after being abandoned and then thrust into a world full of Titans, he'd have to be more cautious. He might not be as open and trusting as he was in his original world. But over time, his natural determination and will to protect others would still shine through as he learns to fight Titans.
Luffy from 'One Piece' would also be an interesting crossover character. Imagine Ichigo being abandoned on an island in the 'One Piece' world. Luffy's carefree yet determined nature could contrast with Ichigo's more brooding personality. They could embark on a journey together, with Ichigo learning about the concept of the 'One Piece' and using his powers in unexpected ways within this new world.
Perhaps it could be a new character created specifically for the fanfiction. A young ninja from a minor clan who showed extraordinary talent in a unique form of jutsu. But because their clan was small and not very influential, and the larger clans felt threatened by this new talent, they managed to get the village to abandon this prodigy.
Another reason might be that they faced some sort of writer's block. Coming up with new plot ideas, especially for a crossover as complex as Naruto and Devil May Cry, can be really challenging. They might have hit a point where they didn't know how to progress the story further. Or perhaps they received some negative feedback early on that discouraged them from continuing.
An interesting concept could be Ichigo crossing over into a fantasy world and being abandoned in a forest filled with magical creatures. He has to quickly adapt, learn about the new magic system, and fight against the prejudices of the native creatures who see him as an outsider. His journey could involve finding a way back home while also protecting this new world from a hidden threat.